Indian Institute of Technology Palakkad&nbsp;(IIT Palakkad or IIT PKD) is a public autonomous&nbsp;engineering&nbsp;and&nbsp;research&nbsp;institute located in&nbsp;Palakkad,&nbsp;Kerala. It was&nbsp;one of the five new&nbsp;IITs&nbsp;proposed in the&nbsp;2014 Union budget of India. The campus was inaugurated on 3 August 2015. The institute has 94 faculty, 978 students, and 63 non-teaching staffs.

IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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IIT Palakkad

Approximate multidimensional Riemann solvers are essential building blocks in designing globally constraint-preserving finite volume time domain and discontinuous Galerkin time domain schemes for computational electrodynamics (CED). In those schemes, we can achieve high-order temporal accuracy with the help of Runge&ndash;Kutta or ADER time-stepping. This paper presents the design of a multidimensional approximate generalized Riemann problem (GRP) solver for the first time. The multidimensional Riemann solver accepts as its inputs the four states surrounding an edge on a structured mesh, and its output consists of a resolved state and its associated fluxes. In contrast, the multidimensional GRP solver accepts as its inputs the four states and their gradients in all directions; its output consists of the resolved state and its corresponding fluxes and the gradients of the resolved state. The gradients can then be used to extend the solution in time. As a result, we achieve second-order temporal accuracy in a single step. In this work, the formulation is optimized for linear hyperbolic systems with stiff, linear source terms because such a formulation will find maximal use in CED. Our formulation produces an overall constraint-preserving time-stepping strategy based on the GRP that is provably L-stable in the presence of stiff source terms. We present several stringent test problems, showing that the multidimensional GRP solver for CED meets its design accuracy and performs stably with optimal time steps. The test problems include cases with high conductivity, showing that the beneficial L-stability is indeed realized in practical applications. &copy; 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

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Journal of Scientific Computing

Multidimensional Generalized Riemann Problem Solver for Maxwell’s Equations

Deep neural network (DNN)-based methods are suitable for the rapid inversion of borehole resistivity measurements. They approximate the forward and the inverse problem offline during the training phase, and they only require a fraction of a second for the online evaluation (aka prediction). Herein, we have adopted a DNN-based iterative algorithm to design a borehole instrument such that the inverse solution is unique for a given earth parameterization. We select a large set of electromagnetic measurement systems routinely used in logging operations, and our DNN algorithm selects a subset of measurements that are suitable for inversion purposes. Numerical results with synthetic data confirm that this approach can provide valuable insight when designing borehole-logging instruments. © 2022 Society of Exploration Geophysicists.

Geophysics

A deep learning approach to design a borehole instrument for geosteering

Discontinuous Galerkin (DG) methods have become mainstays in the accurate solution of hyperbolic systems, which suggests that they should also be important for computational electrodynamics (CED). Typically DG schemes are coupled with Runge-Kutta timestepping, resulting in RKDG schemes, which are also sometimes called DGTD schemes in the CED community. However, Maxwell's equations, which are solved in CED codes, have global mimetic constraints. In Balsara and Käppeli [von Neumann Stability Analysis of Globally Constraint-Preserving DGTD and PNPM Schemes for the Maxwell Equations using Multidimensional Riemann Solvers, Journal of Computational Physics, 376 (2019) 1108-1137] the authors presented globally constraint-preserving DGTD schemes for CED. The resulting schemes had excellent low dissipation and low dispersion properties. Their one deficiency was that the maximal permissible CFL of DGTD schemes decreased with increasing order of accuracy. The goal of this paper is to show how this deficiency is overcome. Because CED entails the propagation of electromagnetic waves, we would also like to obtain DG schemes for CED that minimize dissipation and dispersion errors even more than the prior generation of DGTD schemes. Two recent advances make this possible. The first advance, which has been reported elsewhere, is the development of a multidimensional Generalized Riemann Problem (GRP) solver. The second advance relates to the use of Two Derivative Runge Kutta (TDRK) timestepping. This timestepping uses not just the solution of the multidimensional Riemann problem, it also uses the solution of the multidimensional GRP. When these two advances are melded together, we arrive at DG(TD)2 schemes for CED, where the first “TD” stands for time-derivative and the second “TD” stands for the TDRK timestepping. The first goal of this paper is to show how DG(TD)2 schemes for CED can be formulated with the help of the multidimensional GRP and TDRK timestepping. The second goal of this paper is to utilize the free parameters in TDRK timestepping to arrive at DG(TD)2 schemes for CED that offer a uniformly large CFL with increasing order of accuracy while minimizing the dissipation and dispersion errors to exceptionally low values. The third goal of this paper is to document a von Neumann stability analysis of DG(TD)2 schemes so that their dissipation and dispersion properties can be quantified and studied in detail. At second order we find a DG(TD)2 scheme with CFL of 0.25 and improved dissipation and dispersion properties; for a second order scheme. At third order we present a novel DG(TD)2 scheme with CFL of 0.2571 and improved dissipation and dispersion properties; for a third order scheme. At fourth order we find a DG(TD)2 scheme with CFL of 0.2322 and improved dissipation and dispersion properties. As an extra benefit, the resulting DG(TD)2 schemes for CED require fewer synchronization steps on parallel supercomputers than comparable DGTD schemes for CED. We also document some test problems to show that the methods achieve their design accuracy. © 2020 Elsevier Inc.

Journal of Computational Physics

Optimal, globally constraint-preserving, DG(TD)2 schemes for computational electrodynamics based on two-derivative Runge-Kutta timestepping and multidimensional generalized Riemann problem solvers – A von Neumann stability analysis

Computational electrodynamics (CED), the numerical solution of Maxwell's equations, plays an incredibly important role in several problems in science and engineering. High accuracy solutions are desired, and the discontinuous Galerkin (DG) method is one of the better ways of delivering high accuracy in numerical CED. Maxwell's equations have a pair of involution constraints for which mimetic schemes that globally satisfy the constraints at a discrete level are highly desirable. Balsara and Käppeli (2019) presented a von Neumann stability analysis of globally constraint-preserving DG schemes for CED up to fourth order. That paper was focused on developing the theory and documenting the superior dissipation and dispersion of DGTD schemes in media with constant permittivity and permeability. In this paper we present working DGTD schemes for CED that go up to fifth order of accuracy and analyze their performance when permittivity and permeability vary strongly in space. Our DGTD schemes achieve constraint preservation by collocating the electric displacement and magnetic induction as well as their higher order modes in the faces of the mesh. Our first finding is that at fourth and higher orders of accuracy, one has to evolve some zone-centered modes in addition to the face-centered modes. It is well-known that the limiting step in DG schemes causes a reduction of the optimal accuracy of the scheme; though the schemes still retain their formal order of accuracy with WENO-type limiters. In this paper, we document simulations where permittivity and permeability vary by almost an order of magnitude without requiring any limiting of the DG scheme. This very favorable second finding ensures that DGTD schemes retain optimal accuracy even in the presence of large spatial variations in permittivity and permeability. We also study the conservation of electromagnetic energy in these problems. Our third finding shows that the electromagnetic energy is conserved very well even when permittivity and permeability vary strongly in space; as long as the conductivity is zero. © 2019 Elsevier Inc.

Globally constraint-preserving FR/DG scheme for Maxwell's equations at all orders

Magnetic Resonance Imaging (MRI) is a widely applied non-invasive imaging modality based on non-ionizing radiation which gives excellent images and soft tissue contrast of living tissues. We consider the modified Bloch problem as a model of MRI for flowing spins in an incompressible flow field. After establishing the well-posedness of the corresponding evolution problem, we analyze its spatial semi-discretization using discontinuous Galerkin methods. The high frequency time evolution requires a proper explicit and adaptive temporal discretization. The applicability of the approach is shown for basic examples. © 2017 IMACS